Abstract

Industrial applications frequently pose a notorious challenge for state-of-the-art methods in the contexts of optimization, designing experiments and modeling unknown physical response. This problem is aggravated by limited availability of clean data, uncertainty in available physics-based models and additional logistic and computational expense associated with experiments. In such a scenario, Bayesian methods have played an impactful role in alleviating the aforementioned obstacles by quantifying uncertainty of different types under limited resources. These methods, usually deployed as a framework, allows decision makers to make informed choices under uncertainty while being able to incorporate information on the fly, usually in the form of data, from multiple sources while being consistent with the physical intuition about the problem. This is a major advantage that Bayesian methods bring to fruition especially in the industrial context. This paper is a compendium of the Bayesian modeling methodology that is being consistently developed at GE Research. The methodology, called GE's Bayesian hybrid modeling (GEBHM), is a probabilistic modeling method, based on the Kennedy and O'Hagan framework, that has been continuously scaled-up and industrialized over several years. In this work, we explain the various advancements in GEBHM's methods and demonstrate their impact on several challenging industrial problems.

References

1.
Oden
,
J. T.
,
Moser
,
R.
, and
Ghattas
,
O.
,
2010
, “
Computer Predictions With Quantified Uncertainty: Part II
,”
SIAM News
,
43
(
10
), pp.
1
4
.https://archive.siam.org/news/news.php?id=1857
2.
Ghosh
,
S.
,
Kristensen
,
J.
,
Zhang
,
Y.
,
Subber
,
W.
, and
Wang
,
L.
,
2019
, “
A Strategy for Adaptive Sampling of Multi-Fidelity Gaussian Process to Reduce Predictive Uncertainty
,”
ASME
Paper No. DETC2019-98418.10.1115/DETC2019-98418
3.
Kristensen
,
J.
,
Subber
,
W.
,
Zhang
,
Y.
,
Ghosh
,
S.
,
Kumar
,
N. C.
,
Khan
,
G.
, and
Wang
,
L.
,
2019
, “
Industrial Applications of Intelligent Adaptive Sampling Methods for Multi-Objective Optimization
,”
Design Engineering and Manufacturing
,
IntechOpen
, Rijeka, Croatia.
4.
Santner
,
T. J.
,
Williams
,
B. J.
,
Notz
,
W.
, and
Williams
,
B. J.
,
2003
,
The Design and Analysis of Computer Experiments
, Vol.
1
,
Springer
, Berlin.
5.
Hill
,
M. C.
,
2000
, “
Methods and Guidelines for Effective Model Calibration
,”
Building Partnerships
, Joint Conference on Water Resource Engineering and Water Resources Planning and Management, Minneapolis, MN, July 30–Aug. 2, pp.
1
10
.
6.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2001
, “
Bayesian Calibration of Computer Models (With Discussion)
,”
J. R. Stat. Soc. (Ser. B)
,
63
(
3
), pp.
425
464
.
7.
Forrester
,
A. I.
, and
Keane
,
A. J.
,
2009
, “
Recent Advances in Surrogate-Based Optimization
,”
Prog. Aerosp. Sci.
,
45
(
1–3
), pp.
50
79
.10.1016/j.paerosci.2008.11.001
8.
Wang
,
G. G.
, and
Shan
,
S.
,
2007
, “
Review of Metamodeling Techniques in Support of Engineering Design Optimization
,”
ASME J. Mech. Des.
,
129
(
4
), pp.
370
380
.10.1115/1.2429697
9.
Zhang
,
Y.
,
Meeker
,
J.
,
Schutte
,
J.
,
Kim
,
N.
, and
Hafkta
,
R.
,
2016
, “
On Approaches to Combine Experimental Strength and Simulation With Application to Open-Hole-Tension Configuration
,”
Proceedings of the American Society for Composites: Thirty-First Technical Conference
, Williamsburg, VA, Sept.
10.
Hao
,
P.
,
Wang
,
B.
, and
Li
,
G.
,
2012
, “
Surrogate-Based Optimum Design for Stiffened Shells With Adaptive Sampling
,”
AIAA J.
,
50
(
11
), pp.
2389
2407
.10.2514/1.J051522
11.
Shabouei
,
M.
,
Subber
,
W.
,
Williams
,
C. W.
,
Matouš
,
K.
, and
Powers
,
J. M.
,
2019
, “
Chemo-Thermal Model and Gaussian Process Emulator for Combustion Synthesis of ni/al Composites
,”
Combust. Flame
,
207
, pp.
153
170
.10.1016/j.combustflame.2019.05.038
12.
Zhang
,
S.
,
Zhu
,
P.
, and
Chen
,
W.
,
2013
, “
Crashworthiness-Based Lightweight Design Problem Via New Robust Design Method Considering Two Sources of Uncertainties
,”
Proc. Inst. Mech. Eng., Part C
,
227
(
7
), pp.
1381
1391
.10.1177/0954406212460824
13.
Chaudhuri
,
A.
,
Haftka
,
R. T.
,
Ifju
,
P.
,
Chang
,
K.
,
Tyler
,
C.
, and
Schmitz
,
T.
,
2015
, “
Experimental Flapping Wing Optimization and Uncertainty Quantification Using Limited Samples
,”
Struct. Multidiscip. Optim.
,
51
(
4
), pp.
957
970
.10.1007/s00158-014-1184-x
14.
Hu
,
Z.
, and
Mahadevan
,
S.
,
2016
, “
A Single-Loop Kriging Surrogate Modeling for Time-Dependent Reliability Analysis
,”
ASME J. Mech. Des.
,
138
(
6
), p.
061406
.10.1115/1.4033428
15.
Tapia
,
G.
,
Elwany
,
A.
, and
Sang
,
H.
,
2016
, “
Prediction of Porosity in Metal-Based Additive Manufacturing Using Spatial Gaussian Process Models
,”
Addit. Manuf.
,
12
, pp.
282
290
.10.1016/j.addma.2016.05.009
16.
Higdon
,
D.
,
Nakhleh
,
C.
,
Gattiker
,
J.
, and
Williams
,
B.
,
2008
, “
A Bayesian Calibration Approach to the Thermal Problem
,”
Comput. Methods Appl. Mech. Eng.
,
197
(
29–32
), pp.
2431
2441
.10.1016/j.cma.2007.05.031
17.
Gattiker
,
J.
,
Myers
,
K.
,
Williams
,
B. J.
,
Higdon
,
D.
,
Carzolio
,
M.
, and
Hoegh
,
A.
,
2017
, “
Gaussian Process-Based Sensitivity Analysis and Bayesian Model Calibration With GPMSA
,”
Handbook of Uncertainty Quantification
, R. Ghanem, D. Higdon, and H. Owhadi, eds., Springer, Cham, Switzerland, pp.
1
41
.
18.
Wang
,
L.
,
Fang
,
X.
,
Subramaniyan
,
A.
,
Jothiprasad
,
G.
,
Gardner
,
M.
,
Kale
,
A.
,
Akkaram
,
S.
,
Beeson
,
D.
,
Wiggs
,
G.
, and
Nelson
,
J.
,
2011
, “
Challenges in Uncertainty, Calibration, Validation and Predictability of Engineering Analysis Models
,”
ASME
Paper No. GT2011-46554.10.1115/GT2011-46554
19.
Subramaniyan
,
A. K.
,
Kumar
,
N. C.
,
Wang
,
L.
,
Beeson
,
D.
, and
Wiggs
,
G.
,
2012
, “
Enhancing High-Dimensional Physics Models for Accurate Predictions With Bayesian Calibration
,”
Propulsion-Safety and Affordable Readiness Conference
, Jacksonville, FL, Mar., pp.
20
22
.
20.
Kennedy
,
M. C.
, and
O'Hagan
,
A.
,
2001
, “
Bayesian Calibration of Computer Models
,”
J. R. Stat. Soc. Ser. B (Stat. Methodol.)
,
63
(
3
), pp.
425
464
.10.1111/1467-9868.00294
21.
Poudou
,
O.
, and
Pierre
,
C.
,
2003
, “
Hybrid Frequency-Time Domain Methods for the Analysis of Complex Structural Systems With Dry Friction Damping
,”
AIAA
Paper No. 2003-1411
.10.2514/6.2003-1411
22.
Kumar
,
N. C.
,
Subramaniyan
,
A. K.
,
Wang
,
L.
, and
Wiggs
,
G.
,
2013
, “
Calibrating Transient Models With Multiple Responses Using Bayesian Inverse Techniques
,”
ASME
Paper No. GT2013-95857.10.1115/GT2013-95857
23.
Quinonero-Candela
,
J.
,
Girard
,
A.
, and
Rasmussen
,
C. E.
,
2003
, “
Prediction at an Uncertain Input for Gaussian Processes and Relevance Vector Machines–Application to Multiple-Step Ahead Time-Series Forecasting
,” Technical University of Denmark, Kongens Lyngby, Denmark,
Report
.http://quinonero.net/Publications/QuiGirRas03.pdf
24.
Quinonero-Candela
,
J.
,
Girard
,
A.
,
Larsen
,
J.
, and
Rasmussen
,
C. E.
,
2003
, “
Propagation of Uncertainty in Bayesian Kernels Models—Application to Multiple-Step Ahead Forecasting
,”
International Conference on Acoustics, Speech and Signal Processing
, Hong Kong, China, Apr.
6
10
.10.1109/ICASSP.2003.1202463
25.
Girard
,
A.
,
2004
, “
Approximate Methods for Propagation of Uncertainty With Gaussian Process Models
,” Ph.D. thesis,
University of Glasgow
, Glasgow, UK.
26.
Crespo
,
L.
,
Kenny
,
S.
, and
Giesy
,
D.
,
2014
, “
The Nasa Langley Multidisciplinary Uncertainty Quantification Challenge
,” 16th
AIAA Non-Deterministic Approaches Conference
, National Harbor, MD, Jan. 13–17.https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20140007349.pdf
27.
Ryan
,
K. M.
,
Kristensen
,
J.
,
Ling
,
Y.
,
Ghosh
,
S.
,
Asher
,
I.
, and
Wang
,
L.
,
2018
, “
A Gaussian Process Modeling Approach for Fast Robust Design With Uncertain Inputs
,”
ASME
Paper No. GT2018-77007.10.1115/GT2018-77007
28.
Ghosh
,
S.
,
Pandita
,
P.
,
Subber
,
W.
,
Zhang
,
Y.
, and
Wang
,
L.
,
2020
, “
Efficient Bayesian Inverse Method Using Robust Gaussian Processes for Design Under Uncertainty
,”
AIAA
Paper No.
2020
1877
.10.2514/6.2020-1877
29.
Sobol
,
I. M.
,
2001
, “
Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates
,”
Math. Comput. Simul.
,
55
(
1–3
), pp.
271
280
.10.1016/S0378-4754(00)00270-6
30.
Oakley
,
J. E.
, and
O'Hagan
,
A.
,
2004
, “
Probabilistic Sensitivity Analysis of Complex Models: A Bayesian Approach
,”
J. R. Stat. Soc. Ser. B (Stat. Methodol.)
,
66
(
3
), pp.
751
769
.10.1111/j.1467-9868.2004.05304.x
31.
Srivastava
,
A.
,
Subramaniyan
,
A. K.
, and
Wang
,
L.
,
2017
, “
Analytical Global Sensitivity Analysis With Gaussian Processes
,”
AI Edam
,
31
(
3
), pp.
235
250
.10.1017/S0890060417000142
32.
Saltelli
,
A.
, and
Tarantola
,
S.
,
2002
, “
On the Relative Importance of Input Factors in Mathematical Models: Safety Assessment for Nuclear Waste Disposal
,”
J. Am. Stat. Assoc.
,
97
(
459
), pp.
702
709
.10.1198/016214502388618447
33.
Xu
,
C.
, and
Gertner
,
G. Z.
,
2008
, “
Uncertainty and Sensitivity Analysis for Models With Correlated Parameters
,”
Reliab. Eng. Syst. Saf.
,
93
(
10
), pp.
1563
1573
.10.1016/j.ress.2007.06.003
34.
Li
,
G.
,
Rabitz
,
H.
,
Yelvington
,
P. E.
,
Oluwole
,
O. O.
,
Bacon
,
F.
,
Kolb
,
C. E.
, and
Schoendorf
,
J.
,
2010
, “
Global Sensitivity Analysis for Systems With Independent and/or Correlated Inputs
,”
J. Phys. Chem. A
,
114
(
19
), pp.
6022
6032
.10.1021/jp9096919
35.
Chastaing
,
G.
,
Gamboa
,
F.
, and
Prieur
,
C.
,
2015
, “
Generalized Sobol Sensitivity Indices for Dependent Variables: Numerical Methods
,”
J. Stat. Comput. Simul.
,
85
(
7
), pp.
1306
1333
.10.1080/00949655.2014.960415
36.
Sudret
,
B.
,
2008
, “
Global Sensitivity Analysis Using Polynomial Chaos Expansions
,”
Reliab. Eng. Syst. Saf.
,
93
(
7
), pp.
964
979
.10.1016/j.ress.2007.04.002
37.
Li
,
G.
, and
Rabitz
,
H.
,
2012
, “
General Formulation of HDMR Component Functions With Independent and Correlated Variables
,”
J. Math. Chem.
,
50
(
1
), pp.
99
130
.10.1007/s10910-011-9898-0
38.
Srivastava
,
A.
,
Subramaniyan
,
A. K.
, and
Wang
,
L.
,
2015
, “
Hybrid Bayesian Solution to Nasa Langley Research Center Multidisciplinary Uncertainty Quantification Challenge
,”
J. Aerosp. Inf. Syst.
,
12
(
1
), pp.
114
139
.10.2514/1.I010266
39.
Srivastava
,
A.
,
Subramaniyan
,
A. K.
, and
Wang
,
L.
,
2015
, “
Variance Based Global Sensitivity Analysis for Uncorrelated and Correlated Inputs With Gaussian Processes
,”
ASME
Paper No. GT2015-43693.10.1115/GT2015-43693
40.
Kristensen
,
J.
,
Asher
,
I.
, and
Wang
,
L.
,
2018
, “
Polynomial Representation of the Gaussian Process
,”
ASME
Paper No. DETC2018-85145.10.1115/DETC2018-85145
41.
Ghosh
,
S.
,
Asher
,
I.
,
Kristensen
,
J.
,
Ling
,
Y.
,
Ryan
,
K.
, and
Wang
,
L.
,
2018
, “
Bayesian Multi-Source Modeling With Legacy Data
,” AIAA Non-Deterministic Approaches Conference, Kissimmee, FL, Jan.
42.
Pandita
,
P.
,
Kristensen
,
J.
, and
Wang
,
L.
,
2019
, “
Towards Scalable Gaussian Process Modeling
,” arXiv preprint arXiv:1907.11313.
43.
Bilionis
,
I.
,
Drewniak
,
B. A.
, and
Constantinescu
,
E. M.
,
2015
, “
Crop Physiology Calibration in the CLM
,”
Geosci. Model Develop.
,
8
(
4
), pp.
1071
1083
.10.5194/gmd-8-1071-2015
44.
Ling
,
Y.
,
Ghosh
,
S.
,
Kristensen
,
J.
,
Asher
,
I.
,
Ryan
,
K.
, and
Wang
,
L.
,
2018
, “An
Intelligent Sampling Framework for Multi-Objective Optimization in High Dimensional Design Space
,”
AIAA
Paper No. 2018-0912
.10.2514/6.2018-0912
45.
Kristensen
,
J.
,
Ling
,
Y.
,
Asher
,
I.
, and
Wang
,
L.
,
2016
, “
Expected-Improvement-Based Methods for Adaptive Sampling in Multi-Objective Optimization Problems
,”
ASME
Paper No. DETC2016-59266.10.1115/DETC2016-59266
46.
Subber
,
W.
, and
Loisel
,
S.
,
2014
, “
Schwarz Preconditioners for Stochastic Elliptic PDEs
,”
Comput. Methods Appl. Mech. Eng.
,
272
, pp.
34
57
.10.1016/j.cma.2013.12.016
47.
Subber
,
W.
, and
Sarkar
,
A.
,
2013
, “
Dual-Primal Domain Decomposition Method for Uncertainty Quantification
,”
Comput. Methods Appl. Mech. Eng.
,
266
, pp.
112
124
.10.1016/j.cma.2013.07.007
48.
Subber
,
W.
, and
Sarkar
,
A.
,
2014
, “
A Domain Decomposition Method of Stochastic PDEs: An Iterative Solution Techniques Using a Two-Level Scalable Preconditioner
,”
J. Comput. Phys.
,
257
, pp.
298
317
.10.1016/j.jcp.2013.08.058
49.
Zhang
,
Y.
,
Kristensen
,
J.
,
Subber
,
W.
,
Ghosh
,
S.
,
Khan
,
G.
, and
Wang
,
L.
,
2020
, “
Remarks for Scaling Up a General Gaussian Process to Model Large Dataset With Sub-Models
,”
AIAA
Paper No.
2020
0678
.10.2514/6.2020-0678
50.
Wilson
,
A. G.
,
Hu
,
Z.
,
Salakhutdinov
,
R.
, and
Xing
,
E. P.
,
2016
, “
Deep Kernel Learning
,” 19th International Conference on
Artificial Intelligence and Statistics
, Cadiz, Spain, pp.
370
378
. http://proceedings.mlr.press/v51/wilson16.pdf
51.
Damianou
,
A.
, and
Lawrence
,
N.
,
2013
, “
Deep Gaussian Processes
,”
Proceedings of Machine Learning Research, Scottsdale, AZ, Apr. 29–May 1, pp.
207
215
.
52.
Titsias
,
M.
,
2009
, “
Variational Learning of Inducing Variables in Sparse Gaussian Processes
,”
Proceedings of Machine Learning Research, Clearwater Beach, FL, Apr. 16–19, pp.
567
574
.
53.
Titsias
,
M.
, and
Lázaro-Gredilla
,
M.
,
2014
, “
Doubly Stochastic Variational Bayes for Non-Conjugate Inference
,”
Beijing, China, June 22–24, pp.
1971
1979
.
54.
Salimbeni
,
H.
, and
Deisenroth
,
M.
,
2017
, “
Doubly Stochastic Variational Inference for Deep Gaussian Processes
,”
Advances in Neural Information Processing Systems
, Long Beach, CA, Dec. 4–9, pp.
4588
4599
.
55.
Abadi
,
M.
,
Barham
,
P.
,
Chen
,
J.
,
Chen
,
Z.
,
Davis
,
A.
,
Dean
,
J.
,
Devin
,
M.
,
Ghemawat
,
S.
,
Irving
,
G.
, and
Isard
,
M.
,
2016
, “
Tensorflow: A System for Large-Scale Machine Learning
,”
12th {USENIX} Symposium on Operating Systems Design and Implementatio ({OSDI} 16)
, Savannah, GA, Nov. 2–4, pp.
265
283
.
56.
Paszke
,
A.
,
Gross
,
S.
,
Chintala
,
S.
,
Chanan
,
G.
,
Yang
,
E.
,
DeVito
,
Z.
,
Lin
,
Z.
,
Desmaison
,
A.
,
Antiga
,
L.
, and
Lerer
,
A.
,
2017
, “
Automatic Differentiation in Pytorch
,” Autodiff Workshop in Thirty-First Conference on Neural Information Processing Systems, Long Beach, CA, Dec.
57.
Matthews
,
D. G.
,
Alexander
,
G.
,
Van Der Wilk
,
M.
,
Nickson
,
T.
,
Fujii
,
K.
,
Boukouvalas
,
A.
,
León-Villagrá
,
P.
,
Ghahramani
,
Z.
, and
Hensman
,
J.
,
2017
, “
Gpflow: A Gaussian Process Library Using Tensorflow
,”
J. Mach. Learn. Res.
,
18
(
1
), pp.
1299
1304
.http://jmlr.org/papers/v18/16-537.html
You do not currently have access to this content.